Let [ t ] denote the greatest integer ≤ t . Then 2 π ∫ π 6 5 π 6 8 cosec x - 5 cot x d x is equal to _ _ _ _ _ _ _

Step-by-step solution

Let, I = ∫ π 6 5 π 6 8 cosec x - 5 cot x d x ⇒ I = 8   ∫ π 6 5 π 6 cosec x d x - 5 ∫ π 6 5 π 6 cot x d x ⇒ I = 8 I 1 - 5 I 2 , where I 1 = 8   ∫ π 6 5 π 6 cosec x d x   &   I 2 = 5 ∫ π 6 5 π 6 cot x d x Now solving, I 1 = ∫ π 6 5 π 6 cosec x d x ⇒ I 1 = ∫ π 6 5 π 6 1 d x = 2 π 3 As when x ∈ π 6 , 5 π 6 ,   cosec x ∈ [ 1 , 2 ) , So,